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## Homework Statement

Particle bound by

[tex] V = \frac{1}{2} m \omega^2 x^2 - a x^3 [/tex]

for small x. Show that the mean position of the particle changes with the energy of the eigenstates when [tex]a[/tex] is small, so first order perturbation theory works.

## Homework Equations

For the harmonic oscillator

[tex] x = \sqrt{\frac{\hbar}{2m\omega}}(a^{\dagger}+a) [/tex]

## The Attempt at a Solution

That x^3 perturbation will give an odd number of creation/destruction operators, so there's no shift in energy eigenvalues to first order in perturbation theory. But how does that help answering the question?